Problem: Factor the following expression: $x^2 - 6x - 27$
When we factor a polynomial, we are basically reversing this process of multiplying linear expressions together: $ \begin{eqnarray} (x + a)(x + b) &=& xx &+& xb + ax &+& ab \\ \\ &=& x^2 &+& {(a + b)}x &+& {ab} \end{eqnarray} $ $ \begin{eqnarray} \hphantom{(x + a)(x + b) }&\hphantom{=}&\hphantom{ xx }&\hphantom{+}&\hphantom{ (a + b)x }&\hphantom{+}& \\ &=& x^2 & & {-6}x& & {-27} \end{eqnarray} $ The coefficient on the $x$ term is $-6$ and the constant term is $-27$ , so to reverse the steps above, we need to find two numbers that add up to $-6$ and multiply to $-27$ You can try out different factors of $-27$ to see if you can find two that satisfy both conditions. If you're stuck and can't think of any, you can also rewrite the conditions as a system of equations and try solving for $a$ and $b$ $ {a} + {b} = {-6}$ $ {a} \times {b} = {-27}$ The two numbers $-9$ and $3$ satisfy both conditions: $ {-9} + {3} = {-6} $ $ {-9} \times {3} = {-27} $ So we can factor the expression as: $(x {-9})(x + {3})$